Ergodicity of asymmetric lemon Billiards

Abstract

We show ergodicity of (asymmetric) lemon billiards, billiard tables that are the intersection of two circles of which one contains the centers of both. These do not satisfy the Wojtkowski criteria for hyperbolicity, but we establish \emph{uniform} expansion of vectors in an invariant cone family and alignment of singularity curves. Both of these are difficult, and the approach to the latter seems new. Together, these properties imply ergodicity.